Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,Mod(32,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.32"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-360,0,0,9288] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{86})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1849 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 32.2
Root \(4.63681 - 4.63681i\) of defining polynomial
Character \(\chi\) \(=\) 75.32
Dual form 75.10.e.a.68.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(27.8209 - 27.8209i) q^{2} +(-48.2687 + 131.731i) q^{3} -1036.00i q^{4} +(2322.00 + 5007.75i) q^{6} +(6300.00 + 6300.00i) q^{7} +(-14578.1 - 14578.1i) q^{8} +(-15023.3 - 12717.0i) q^{9} -10015.5i q^{11} +(136474. + 50006.4i) q^{12} +(-113760. + 113760. i) q^{13} +350543. q^{14} -280720. q^{16} +(-418259. + 418259. i) q^{17} +(-771758. + 64162.2i) q^{18} +74396.0i q^{19} +(-1.13400e6 + 525814. i) q^{21} +(-278640. - 278640. i) q^{22} +(1.15298e6 + 1.15298e6i) q^{23} +(2.62406e6 - 1.21673e6i) q^{24} +6.32980e6i q^{26} +(2.40038e6 - 1.36520e6i) q^{27} +(6.52680e6 - 6.52680e6i) q^{28} +5.61870e6 q^{29} -1.72993e6 q^{31} +(-345869. + 345869. i) q^{32} +(1.31936e6 + 483436. i) q^{33} +2.32726e7i q^{34} +(-1.31748e7 + 1.55641e7i) q^{36} +(5.75352e6 + 5.75352e6i) q^{37} +(2.06976e6 + 2.06976e6i) q^{38} +(-9.49470e6 - 2.04768e7i) q^{39} +1.54339e7i q^{41} +(-1.69203e7 + 4.61775e7i) q^{42} +(-2.43693e7 + 2.43693e7i) q^{43} -1.03761e7 q^{44} +6.41538e7 q^{46} +(8.82675e6 - 8.82675e6i) q^{47} +(1.35500e7 - 3.69796e7i) q^{48} +3.90264e7i q^{49} +(-3.49089e7 - 7.52866e7i) q^{51} +(1.17855e8 + 1.17855e8i) q^{52} +(4.01908e7 + 4.01908e7i) q^{53} +(2.87996e7 - 1.04762e8i) q^{54} -1.83684e8i q^{56} +(-9.80028e6 - 3.59100e6i) q^{57} +(1.56317e8 - 1.56317e8i) q^{58} -1.04091e8 q^{59} +1.51862e7 q^{61} +(-4.81281e7 + 4.81281e7i) q^{62} +(-1.45295e7 - 1.74764e8i) q^{63} -1.24484e8i q^{64} +(5.01552e7 - 2.32560e7i) q^{66} +(-5.07649e7 - 5.07649e7i) q^{67} +(4.33316e8 + 4.33316e8i) q^{68} +(-2.07536e8 + 9.62306e7i) q^{69} -4.26861e7i q^{71} +(3.36210e7 + 4.04401e8i) q^{72} +(3.98110e7 - 3.98110e7i) q^{73} +3.20136e8 q^{74} +7.70743e7 q^{76} +(6.30977e7 - 6.30977e7i) q^{77} +(-8.33833e8 - 3.05531e8i) q^{78} -1.98988e8i q^{79} +(6.39763e7 + 3.82102e8i) q^{81} +(4.29384e8 + 4.29384e8i) q^{82} +(-1.82905e7 - 1.82905e7i) q^{83} +(5.44743e8 + 1.17482e9i) q^{84} +1.35595e9i q^{86} +(-2.71207e8 + 7.40159e8i) q^{87} +(-1.46007e8 + 1.46007e8i) q^{88} -7.56792e8 q^{89} -1.43338e9 q^{91} +(1.19449e9 - 1.19449e9i) q^{92} +(8.35014e7 - 2.27886e8i) q^{93} -4.91136e8i q^{94} +(-2.88671e7 - 6.22564e7i) q^{96} +(-8.05534e8 - 8.05534e8i) q^{97} +(1.08575e9 + 1.08575e9i) q^{98} +(-1.27367e8 + 1.50466e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 360 q^{3} + 9288 q^{6} + 25200 q^{7} + 372960 q^{12} - 455040 q^{13} - 1122880 q^{16} - 1671840 q^{18} - 4536000 q^{21} - 1114560 q^{22} + 2070360 q^{27} + 26107200 q^{28} - 6919712 q^{31} + 1671840 q^{33}+ \cdots - 3222135360 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 27.8209 − 27.8209i 1.22952 − 1.22952i 0.265374 − 0.964145i \(-0.414504\pi\)
0.964145 − 0.265374i \(-0.0854955\pi\)
\(3\) −48.2687 + 131.731i −0.344049 + 0.938952i
\(4\) − 1036.00i − 2.02344i
\(5\) 0 0
\(6\) 2322.00 + 5007.75i 0.731445 + 1.57747i
\(7\) 6300.00 + 6300.00i 0.991744 + 0.991744i 0.999966 − 0.00822268i \(-0.00261739\pi\)
−0.00822268 + 0.999966i \(0.502617\pi\)
\(8\) −14578.1 − 14578.1i −1.25834 − 1.25834i
\(9\) −15023.3 − 12717.0i −0.763261 − 0.646091i
\(10\) 0 0
\(11\) − 10015.5i − 0.206256i −0.994668 − 0.103128i \(-0.967115\pi\)
0.994668 − 0.103128i \(-0.0328851\pi\)
\(12\) 136474. + 50006.4i 1.89991 + 0.696161i
\(13\) −113760. + 113760.i −1.10470 + 1.10470i −0.110865 + 0.993835i \(0.535362\pi\)
−0.993835 + 0.110865i \(0.964638\pi\)
\(14\) 350543. 2.43874
\(15\) 0 0
\(16\) −280720. −1.07086
\(17\) −418259. + 418259.i −1.21458 + 1.21458i −0.245072 + 0.969505i \(0.578811\pi\)
−0.969505 + 0.245072i \(0.921189\pi\)
\(18\) −771758. + 64162.2i −1.73283 + 0.144063i
\(19\) 74396.0i 0.130966i 0.997854 + 0.0654830i \(0.0208588\pi\)
−0.997854 + 0.0654830i \(0.979141\pi\)
\(20\) 0 0
\(21\) −1.13400e6 + 525814.i −1.27241 + 0.589991i
\(22\) −278640. − 278640.i −0.253595 − 0.253595i
\(23\) 1.15298e6 + 1.15298e6i 0.859105 + 0.859105i 0.991233 − 0.132127i \(-0.0421808\pi\)
−0.132127 + 0.991233i \(0.542181\pi\)
\(24\) 2.62406e6 − 1.21673e6i 1.61445 − 0.748588i
\(25\) 0 0
\(26\) 6.32980e6i 2.71650i
\(27\) 2.40038e6 − 1.36520e6i 0.869247 − 0.494378i
\(28\) 6.52680e6 − 6.52680e6i 2.00673 − 2.00673i
\(29\) 5.61870e6 1.47518 0.737590 − 0.675249i \(-0.235964\pi\)
0.737590 + 0.675249i \(0.235964\pi\)
\(30\) 0 0
\(31\) −1.72993e6 −0.336434 −0.168217 − 0.985750i \(-0.553801\pi\)
−0.168217 + 0.985750i \(0.553801\pi\)
\(32\) −345869. + 345869.i −0.0583091 + 0.0583091i
\(33\) 1.31936e6 + 483436.i 0.193664 + 0.0709620i
\(34\) 2.32726e7i 2.98669i
\(35\) 0 0
\(36\) −1.31748e7 + 1.55641e7i −1.30732 + 1.54441i
\(37\) 5.75352e6 + 5.75352e6i 0.504691 + 0.504691i 0.912892 − 0.408201i \(-0.133844\pi\)
−0.408201 + 0.912892i \(0.633844\pi\)
\(38\) 2.06976e6 + 2.06976e6i 0.161025 + 0.161025i
\(39\) −9.49470e6 − 2.04768e7i −0.657189 − 1.41733i
\(40\) 0 0
\(41\) 1.54339e7i 0.852998i 0.904488 + 0.426499i \(0.140253\pi\)
−0.904488 + 0.426499i \(0.859747\pi\)
\(42\) −1.69203e7 + 4.61775e7i −0.839044 + 2.28986i
\(43\) −2.43693e7 + 2.43693e7i −1.08701 + 1.08701i −0.0911792 + 0.995835i \(0.529064\pi\)
−0.995835 + 0.0911792i \(0.970936\pi\)
\(44\) −1.03761e7 −0.417345
\(45\) 0 0
\(46\) 6.41538e7 2.11257
\(47\) 8.82675e6 − 8.82675e6i 0.263852 − 0.263852i −0.562765 − 0.826617i \(-0.690262\pi\)
0.826617 + 0.562765i \(0.190262\pi\)
\(48\) 1.35500e7 − 3.69796e7i 0.368429 − 1.00549i
\(49\) 3.90264e7i 0.967110i
\(50\) 0 0
\(51\) −3.49089e7 − 7.52866e7i −0.722555 − 1.55830i
\(52\) 1.17855e8 + 1.17855e8i 2.23529 + 2.23529i
\(53\) 4.01908e7 + 4.01908e7i 0.699657 + 0.699657i 0.964336 − 0.264680i \(-0.0852662\pi\)
−0.264680 + 0.964336i \(0.585266\pi\)
\(54\) 2.87996e7 − 1.04762e8i 0.460908 − 1.67660i
\(55\) 0 0
\(56\) − 1.83684e8i − 2.49589i
\(57\) −9.80028e6 − 3.59100e6i −0.122971 − 0.0450587i
\(58\) 1.56317e8 − 1.56317e8i 1.81376 − 1.81376i
\(59\) −1.04091e8 −1.11836 −0.559178 − 0.829048i \(-0.688883\pi\)
−0.559178 + 0.829048i \(0.688883\pi\)
\(60\) 0 0
\(61\) 1.51862e7 0.140432 0.0702160 − 0.997532i \(-0.477631\pi\)
0.0702160 + 0.997532i \(0.477631\pi\)
\(62\) −4.81281e7 + 4.81281e7i −0.413653 + 0.413653i
\(63\) −1.45295e7 − 1.74764e8i −0.116203 − 1.39772i
\(64\) − 1.24484e8i − 0.927477i
\(65\) 0 0
\(66\) 5.01552e7 − 2.32560e7i 0.325363 − 0.150865i
\(67\) −5.07649e7 − 5.07649e7i −0.307770 − 0.307770i 0.536274 − 0.844044i \(-0.319831\pi\)
−0.844044 + 0.536274i \(0.819831\pi\)
\(68\) 4.33316e8 + 4.33316e8i 2.45762 + 2.45762i
\(69\) −2.07536e8 + 9.62306e7i −1.10223 + 0.511084i
\(70\) 0 0
\(71\) − 4.26861e7i − 0.199353i −0.995020 − 0.0996767i \(-0.968219\pi\)
0.995020 − 0.0996767i \(-0.0317809\pi\)
\(72\) 3.36210e7 + 4.04401e8i 0.147440 + 1.77344i
\(73\) 3.98110e7 − 3.98110e7i 0.164078 − 0.164078i −0.620293 − 0.784370i \(-0.712986\pi\)
0.784370 + 0.620293i \(0.212986\pi\)
\(74\) 3.20136e8 1.24106
\(75\) 0 0
\(76\) 7.70743e7 0.265001
\(77\) 6.30977e7 − 6.30977e7i 0.204553 − 0.204553i
\(78\) −8.33833e8 − 3.05531e8i −2.55066 − 0.934609i
\(79\) − 1.98988e8i − 0.574784i −0.957813 − 0.287392i \(-0.907212\pi\)
0.957813 − 0.287392i \(-0.0927882\pi\)
\(80\) 0 0
\(81\) 6.39763e7 + 3.82102e8i 0.165134 + 0.986271i
\(82\) 4.29384e8 + 4.29384e8i 1.04878 + 1.04878i
\(83\) −1.82905e7 − 1.82905e7i −0.0423032 − 0.0423032i 0.685639 − 0.727942i \(-0.259523\pi\)
−0.727942 + 0.685639i \(0.759523\pi\)
\(84\) 5.44743e8 + 1.17482e9i 1.19381 + 2.57464i
\(85\) 0 0
\(86\) 1.35595e9i 2.67301i
\(87\) −2.71207e8 + 7.40159e8i −0.507534 + 1.38512i
\(88\) −1.46007e8 + 1.46007e8i −0.259539 + 0.259539i
\(89\) −7.56792e8 −1.27856 −0.639280 − 0.768974i \(-0.720768\pi\)
−0.639280 + 0.768974i \(0.720768\pi\)
\(90\) 0 0
\(91\) −1.43338e9 −2.19116
\(92\) 1.19449e9 − 1.19449e9i 1.73835 − 1.73835i
\(93\) 8.35014e7 − 2.27886e8i 0.115750 − 0.315896i
\(94\) − 4.91136e8i − 0.648823i
\(95\) 0 0
\(96\) −2.88671e7 − 6.22564e7i −0.0346883 − 0.0748107i
\(97\) −8.05534e8 − 8.05534e8i −0.923870 − 0.923870i 0.0734301 − 0.997300i \(-0.476605\pi\)
−0.997300 + 0.0734301i \(0.976605\pi\)
\(98\) 1.08575e9 + 1.08575e9i 1.18908 + 1.18908i
\(99\) −1.27367e8 + 1.50466e8i −0.133260 + 0.157427i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 75.10.e.a.32.2 yes 4
3.2 odd 2 inner 75.10.e.a.32.1 ✓ 4
5.2 odd 4 75.10.e.d.68.2 yes 4
5.3 odd 4 inner 75.10.e.a.68.1 yes 4
5.4 even 2 75.10.e.d.32.1 yes 4
15.2 even 4 75.10.e.d.68.1 yes 4
15.8 even 4 inner 75.10.e.a.68.2 yes 4
15.14 odd 2 75.10.e.d.32.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.10.e.a.32.1 ✓ 4 3.2 odd 2 inner
75.10.e.a.32.2 yes 4 1.1 even 1 trivial
75.10.e.a.68.1 yes 4 5.3 odd 4 inner
75.10.e.a.68.2 yes 4 15.8 even 4 inner
75.10.e.d.32.1 yes 4 5.4 even 2
75.10.e.d.32.2 yes 4 15.14 odd 2
75.10.e.d.68.1 yes 4 15.2 even 4
75.10.e.d.68.2 yes 4 5.2 odd 4